A Williams number of the second kind base b is a natural number of the form (b−1)⋅bn+1{\displaystyle (b-1)\cdot b^{n}+1} for integers b ≥ 2 and n ≥ 1, a Williams prime of the second kind is a Williams number of the second kind that is prime. The Williams primes of the second kind base 2 are exactly the Fermat primes.

As of September 2018[update], the largest known Williams prime of the second kind base 3 is 2×31175232+1[3].

A Williams number of the third kind base b is a natural number of the form (b+1)⋅bn−1{\displaystyle (b+1)\cdot b^{n}-1} for integers b ≥ 2 and n ≥ 1, the Williams number of the third kind base 2 are exactly the Thabit numbers. A Williams prime of the third kind is a Williams number of the third kind that is prime.

A Williams number of the fourth kind base b is a natural number of the form (b+1)⋅bn+1{\displaystyle (b+1)\cdot b^{n}+1} for integers b ≥ 2 and n ≥ 1, a Williams prime of the fourth kind is a Williams number of the fourth kind that is prime, such primes do not exist for b≡1mod3{\displaystyle b\equiv 1{\bmod {3}}}.

b

numbers n such that (b+1)⋅bn−1{\displaystyle (b+1)\cdot b^{n}-1} is prime

numbers n such that (b+1)⋅bn+1{\displaystyle (b+1)\cdot b^{n}+1} is prime

It is conjectured that for every b ≥ 2, there are infinitely many Williams primes of the first kind (the original Williams primes) base b, infinitely many Williams primes of the second kind base b, and infinitely many Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many Williams primes of the fourth kind base b.

Unlike the original Williams primes of each kind, some large dual Williams primes of each kind are only probable primes, since for these primes N, neither N−1 not N+1 can be trivially written into a product.

b

numbers n such that bn−(b−1){\displaystyle b^{n}-(b-1)} is (probable) prime (dual Williams primes of the first kind)

numbers n such that bn+(b−1){\displaystyle b^{n}+(b-1)} is (probable) prime (dual Williams primes of the second kind)

numbers n such that bn−(b+1){\displaystyle b^{n}-(b+1)} is (probable) prime (dual Williams primes of the third kind)

numbers n such that bn+(b+1){\displaystyle b^{n}+(b+1)} is (probable) prime (dual Williams primes of the fourth kind)

It is conjectured that for every b ≥ 2, there are infinitely many dual Williams primes of the first kind (the original Williams primes) base b, infinitely many dual Williams primes of the second kind base b, and infinitely many dual Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many dual Williams primes of the fourth kind base b.